Unstructured Finite Element Meshes Research Articles

A Spatial Interpolation Approach to Assign Magnetic Resonance Imaging-Derived Material Properties for Finite Element Models of Adeno-Associated Virus Infusion Into a Recurrent Brain Tumor.

Adeno-associated virus (AAV) is a clinically useful gene delivery vehicle for treating neurological diseases. To deliver AAV to focal targets, direct infusion into brain tissue by convection-enhanced delivery (CED) is often needed due to AAV's limited penetration across the blood-brain-barrier and its low diffusivity in tissue. In this study, computational models that predict the spatial distribution of AAV in brain tissue during CED were developed to guide future placement of infusion catheters in recurrent brain tumors following primary tumor resection. The brain was modeled as a porous medium, and material property fields that account for magnetic resonance imaging (MRI)-derived anatomical regions were interpolated and directly assigned to an unstructured finite element mesh. By eliminating the need to mesh complex surfaces between fluid regions and tissue, mesh preparation was expedited, increasing the model's clinical feasibility. The infusion model predicted preferential fluid diversion into open fluid regions such as the ventricles and subarachnoid space (SAS). Additionally, a sensitivity analysis of AAV delivery demonstrated that improved AAV distribution in the tumor was achieved at higher tumor hydraulic conductivity or lower tumor porosity. Depending on the tumor infusion site, the AAV distribution covered 3.67-70.25% of the tumor volume (using a 10% AAV concentration threshold), demonstrating the model's potential to inform the selection of infusion sites for maximal tumor coverage.

Applying Convolutional Neural Networks to data on unstructured meshes with space-filling curves

This paper presents the first classical Convolutional Neural Network (CNN) that can be applied directly to data from unstructured finite element meshes or control volume grids. CNNs have been hugely influential in the areas of image classification and image compression, both of which typically deal with data on structured grids. Unstructured meshes are frequently used to solve partial differential equations and are particularly suitable for problems that require the mesh to conform to complex geometries or for problems that require variable mesh resolution. Central to our approach are space-filling curves, which traverse the nodes or cells of a mesh tracing out a path that is as short as possible (in terms of numbers of edges) and that visits each node or cell exactly once. The space-filling curves (SFCs) are used to find an ordering of the nodes or cells that can transform multi-dimensional solutions on unstructured meshes into a one-dimensional (1D) representation, to which 1D convolutional layers can then be applied. Although developed in two dimensions, the approach is applicable to higher dimensional problems. To demonstrate the approach, the network we choose is a convolutional autoencoder (CAE), although other types of CNN could be used. The approach is tested by applying CAEs to data sets that have been reordered with a space-filling curve. Sparse layers are used at the input and output of the autoencoder, and the use of multiple SFCs is explored. We compare the accuracy of the SFC-based CAE with that of a classical CAE applied to two idealised problems on structured meshes, and then apply the approach to solutions of flow past a cylinder obtained using the finite-element method and an unstructured mesh.

A cut-cell method for the numerical simulation of 3D multiphase flows with strong interfacial effects

We present a numerical model for the approximation of multiphase flows with free surfaces and strong interfacial effects. The model relies on the multiphase incompressible Navier-Stokes equations, and includes surface tension effects on the interfaces between phases, and contact angles. The volume-of-fluid approach is used to track the interfaces and the free surfaces between the various phases and the ambient air. The numerical method relies on an operator splitting strategy. The space discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. An adaptive mesh refinement algorithm is incorporated to better track the interfaces and free surfaces on the finite element mesh, and approximate more accurately surface forces. The model is validated through numerical experiments, in particular for emulsion problems.

A heterogeneous parallel model of unstructured mesh finite element method based on CPU+GPU

Most of the existing numerical simulation programs using the unstructured mesh finite element are based on the traditional multicore processor architecture. With the increase of the number of computing meshes, the computing time is increasing, which leads to the common multicore CPU cluster can’t meet the high computing demand of complex applications. In order to adapt to the trend of the heterogeneous development of high-performance computers, a heterogeneous parallel model of unstructured mesh finite element method is proposed in this paper. It can transplant the unstructured mesh finite element program framework to heterogeneous platform better and faster. The model realizes the efficient utilization of the multicore CPU by hierarchical parallelization, and realizes the efficient utilization of GPU by heterogeneous parallel rewriting for time-consuming computing hotspot. Finally, the model is applied to the parallel transplantation of CPU + GPU heterogeneous platform for the thermal radiation effect program. The results show that the model can reduce the programming difficulty and has good portability and extensibility.

Numerical calculation and fast method for the magnetohydrodynamic flow and heat transfer of fractional Jeffrey fluid on a two-dimensional irregular convex domain

In this paper, we study the magnetohydrodynamic (MHD) flow and heat transfer of the fractional Jeffrey fluid in a straight channel, of which the cross section is a two-dimensional irregular convex domain. We consider a spatial fractional operator to modify the classical Fourier's law of thermal conduction, and obtain the time-space fractional coupled model. The fractional coupled model is solved numerically by the L2−1σ method in the temporal direction and the unstructured mesh finite element method in the spatial direction. Besides, we prove the stability and convergence of the numerical scheme. In order to reduce the computational time, a fast method is proposed. Finally, a numerical example is given to verify the theoretical analysis and the efficiency of the fast method. Furthermore, an example is considered to discuss the effects of the fractional parameters on the MHD flow and heat transfer of the fractional Jeffrey fluid.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

Numerical simulation of temperature-driven free surface flows, with application to laser melting and polishing

We present a multi-physics model for the approximation of the coupled system formed by the heat equation and the Navier-Stokes equations with solidification and free surfaces. The computational domain is the union of two overlapping regions: a larger domain to account for thermal effects, and a smaller region to account for the fluid flow. Temperature-dependent surface effects are accounted for via surface tension and Marangoni forces. The volume-of-fluid approach is used to track the free surfaces between the metal (liquid or solidified) and the ambient air. The numerical method incorporates all the physical phenomena within an operator splitting strategy. The discretization relies on a two-grid approach that uses an unstructured finite element mesh for diffusion phenomena and a structured Cartesian grid for advection phenomena. The model is validated through numerical experiments, the main application being laser melting and polishing.

An interface preserving and residual-based adaptivity for phase-field modeling of fully Eulerian fluid-structure interaction

In this paper, we present a novel interface preserving and residual-based adaptivity procedure for the phase-field modeling of a fully Eulerian fluid-structure interaction. The proposed adaptive fully Eulerian procedure involves a fixed background unstructured finite element mesh on which the fluid-structure boundary is treated implicitly via a diffused interface description. We employ the stabilized finite element formulation and a partitioned iterative procedure to solve the coupled system of the Allen-Cahn phase-field equation with the unified momentum equation comprising solid and fluid dynamics. To evaluate the solid stresses, the left Cauchy-Green deformation tensor is convected at each time step to trace the evolution of the solid strain in the Eulerian reference frame. We utilize a residual based error indicator and the newest vertex bisection algorithm for the adaptive refinement/coarsening of the unstructured mesh. Through the adaptive finite element method, we aim to preserve the profile of the diffused interface in the presence of convective distortion. We propose an early stopping criterion for the adaptivity of the interface dynamics by combining the effects of the phase-field interface thickness parameter and the mesh resolution. In addition, we control the size of the refined/coarsened elements by introducing a characteristic length scale governed by a weighted harmonic mean of the desired bulk and the interface mesh resolutions. The proposed stopping criterion and the characteristic length scale provide significant reductions in the computational cost while simultaneously ensuring convergence properties of the coupled fluid-structure system. We perform a detailed convergence and accuracy analysis via two benchmark problems namely, the pure solid system and a coupled fluid-solid system with an interface in a rectangular domain. We further provide a comparative study between feature- and residual based adaptive strategies. We systematically assess the performance of the adaptive procedure in terms of conservation properties and computational efficiency. Finally, we demonstrate our fully Eulerian adaptive solver to simulate the contact and bouncing phenomenon between an elastic solid and a rigid wall.

A partially mesh-free scheme for representing anisotropic spatial variations along field lines: Conservation, quadrature, and the delta property

A method for representing highly anisotropic fields is presented, based on a partially meshfree Galerkin formulation. A mapping function is used to provide information about the local direction of the anisotropy, with one of the global coordinates chosen to parameterize the ‘parallel’ position along the mapping in a one-to-one manner. Standard unstructured finite element meshes are used on planes of constant parallel coordinate to represent the necessary small-scale variations perpendicular to the mapping direction, with large spacings then possible between these planes because of the large-scale variations along the mapping. This greatly reduces the number of degrees of freedom required to represent fields in this space and the associated computational cost of simulations involving such fields. No mesh connectivity is defined between planes, and field aligned basis functions are constructed using the mapping function to extend the standard finite element basis into the full domain. Integration of the basis has been addressed with reference to methods developed for fully meshfree methods, and the scheme (as well as other similar element-free Galerkin schemes) is shown to be locally conservative under certain conditions. Robust convergence of several test problems is demonstrated.

Dissipation and Bathymetric Sensitivities in an Unstructured Mesh Global Tidal Model

AbstractThe mechanisms and geographic distribution of global tidal dissipation in barotropic tidal models are examined using a high resolution unstructured mesh finite element model. Mesh resolution varies between 2 and 25 km and is especially focused on inner shelves and steep bathymetric gradients. Tidal response sensitivities to bathymetric changes are examined to put into context response sensitivities to frictional processes. We confirm that the Ronne Ice Shelf dramatically affects Atlantic tides but also find that bathymetry in the Hudson Bay system is a critical control. We follow a sequential frictional parameter optimization process and use TPXO9 data‐assimilated tidal elevations as a reference solution. From simulated velocities and depths, dissipation within the global model is estimated and allows us to pinpoint dissipation at high resolution. Boundary layer dissipation is extremely focused with 1.4% of the ocean accounting for 90% of the total. Internal tide friction is much more distributed with 16.7% of the ocean accounting for 90% of the total. Often highly regional dissipation can impact basin‐scale and even ocean wide tides. Optimized boundary layer friction parameters correlate very well with the physical characteristics of the locality with high friction factors associated with energetic tidal regions, deep ocean island chains, and ice covered areas. Global complex M2 tide errors are 1.94 cm in deep waters. Total global boundary layer and internal tide dissipation are estimated, respectively, at 1.83 and 1.49 TW. This continues the trend in the literature toward attributing more dissipation to internal tides.

A Systematic Approach to Constructing Preconditioners for the $hp$-Version Mass Matrix on Unstructured and Hybrid Finite Element Meshes

We develop a systematic approach to building uniform preconditioners for the mass matrix on unstructured meshes composed of higher dimensional elements using only preconditioners for lower dimensional simplices. In particular, we show that the resulting preconditioners are automatically uniform with respect to both the mesh size h and polynomial degree $p$, and that the preconditioners can be implemented efficiently by exploiting the structure of the preconditioners on the lower dimensional elements. We illustrate the approach by developing preconditioners for prismatic elements and for the challenging case of hybrid meshes of hexahedra, prisms, and tetrahedra.

A variational interface-preserving and conservative phase-field method for the surface tension effect in two-phase flows

We present a finite element based variational interface-preserving and conservative phase-field formulation for the modeling of incompressible two-phase flows with surface tension dynamics. The preservation of the hyperbolic tangent interface profile of the convective Allen-Cahn phase-field formulation relies on a novel time-dependent mobility model. The mobility coefficient is adjusted adaptively as a function of gradients of the velocity and the order parameter in the diffuse interface region in such a way that the free energy minimization properly opposes the convective distortion. The ratio of the convective distortion to the free energy minimization is termed as the convective distortion parameter, which characterizes the deviation of the diffuse interface profile from the hyperbolic tangent shape due to the convection effect. In the phase-field formulation, the mass conservation is achieved by enforcing a Lagrange multiplier with both temporal and spatial dependence on the phase-field function. We integrate the interface-preserving and conservative phase-field formulation with the incompressible Navier-Stokes equations and the continuum surface tension force model for the simulation of incompressible two-phase flows. A positivity preserving scheme designed for the boundedness and stability of the solution is employed for the variational discretization using unstructured finite elements. We examine the convergence and accuracy of the Allen-Cahn phase-field solver through a generic one-dimensional bistable convection-diffusion-reaction system in a stretching flow. We quantify and systematically assess the relative interface thickness error and the relative surface tension force error with respect to the convective distortion parameter. Two- and three-dimensional rising bubble cases are further simulated to examine the effectiveness of the proposed model on the volume-preserving mean curvature flow and the interface-preserving capability. Finally, we demonstrate the applicability of the proposed model for a case of two bubbles rising and merging with a free surface in an unstructured finite element mesh, which includes complex topological changes and surface tension dynamics.

Error analysis of the unstructured mesh finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials.

New Algorithm for Three-Dimensional Borehole-to-Surface Apparent Resistivity Imaging Based on Unstructured Mesh Finite-Element Method

In this paper, a finite-element method with unstructured mesh is used to numerically simulate the line source model of three-dimensional borehole-to-surface electrical potential method. A new total field potential is synthesized using the calculated anomalous field potential and the analytic solution of the background field. In solving for the original total field potential and background potential, this method can guarantee a small near-field error in the new total field potential without the need to densely mesh the region with the source. By comparing the synthesized total field and analytic solution, the high accuracy of the numerical solution is verified. The results of model calculation show that the detection effectiveness of the borehole-to-surface electrical method depends on the length of the line source. The detection effectiveness increases substantially with increasing length of the line source within a certain range. However, when the line source is excessively long, the increase in detection effectiveness for anomalies is insignificant. For a low-resistance anomaly of a given volume, the borehole-to-surface electrical method is more sensitive to changes in the lateral area than to changes in the thickness. When the gob contains only a small amount of water (e.g., Model IV containing 5% of water by volume), it will appear as a low-resistance anomaly. In addition, the detection of a water-logged gob by the borehole-to-surface electrical method depends on the depth and size of the gob. The borehole-to-surface method can accurately delineate the profile boundary of an anomaly, even though the side closer to the source is susceptible to the effect of source current. These results are significant for guiding the field exploration of the borehole-to-surface electrical method, improving result interpretation, and assisting the application.

Unstructured finite element mesh decimation for real-time Hurricane storm surge forecasting

A previously developed research-grade (e.g. high-resolution) unstructured mesh of the northern Gulf of Mexico (named NGOM3) is optimized to produce a computationally efficient forecast-grade mesh for deployment in a real-time hurricane storm surge early warning system. The real-time mesh is developed from a mesh decimation scheme with focus on the coastal floodplain. The mesh decimation scheme reduces mesh nodes and elements from the research-grade mesh while preserving the representation of the bare-earth topography. The resulting real-time unstructured mesh (named NGOM-RT) contains 64% less mesh nodes than the research-grade mesh. Comparison of (ADCIRC + SWAN) simulated times-series and peak water levels to observations between the research-grade and real-time-grade meshes for Hurricanes Ivan (2004), Dennis (2005), Katrina (2005), and Isaac (2012) show virtually no difference. Model simulations with the NGOM-RT mesh are 1.5–2.0 times faster than using NGOM3 on the same number of compute cores.

Towards a climate-driven simulation of coupled surface-subsurface hydrology at the continental scale: a Canadian example

The development of new, large-scale tools to evaluate water resources is critical to understanding the long-term sustainability of this resource under future land use, climate change, and population growth. In cold and humid regions it is imperative that such tools consider the hydrologic complexities associated with permafrost and groundwater-surface water (GW-SW) interactions, as these factors are recognized to have significant influence on the global water cycle. In this work we present a physics-based, three-dimensional, fully-integrated GW-SW model for Continental Canada constructed with the HydroGeoSphere simulation platform. The Canadian Continental Basin Model (CCBM) domain, which covers approximately 10.5 million km2, is discretized using an unstructured control-volume finite element mesh that conforms to key river basin boundaries, lakes, and river networks. In order to construct the model, surficial geology maps were assembled, which were combined with near-surface information and bedrock geology into a seven-layer subsurface domain. For the large-scale demonstration, the model was used to simulate historic groundwater levels, surface water flow rates (R2=0.85), and lake levels (R2=0.99) across the domain, with results showing that these targets are well reproduced. To demonstrate the regional-scale utility, simulation results were used to perform a regional groundwater flow analysis for western Canada and a water balance analysis for the Laurentian Great Lakes (Superior, Michigan, Huron, Erie and Ontario). The outcome of this work demonstrates that large-scale fully-integrated hydrologic modeling is possible and can be employed to quantify components of a large-scale water balance that are otherwise difficult or impossible to obtain.

Application of the TELEMAC-2D Model in the Fluvial Hydrodinamics Simulation and Reproduction of Flood Patterns

. The increased occurrence of floods in the city of Rio do Sul (SC), even with the creation of dams to contain floods, show that non-structural measures can be good alternatives to reduce losses in the region. Numerical flood modeling has been widely used to anticipate risks and assist in decisionmaking. One of the numerical models that is being used to simulate floods is TELEMAC-2D, which is able to simulate the hydrodynamics of open channels by solving the shallow water equations in a domain discretized by an unstructured finite element mesh. We used the TELEMAC-2D model tosimulate the dynamics of the rivers of the region of Rio do Sul throughout the year of 2013, period during which a flood with large proportions occurred in September. Fluviometric data avaliable from the National Water Agency and high resolution (1 m) topographic data provided by government agen-cies of Santa Catarina were used in the simulation. The results show that the model performed well in simulating the maximum flood extension occurred in September, however, the simulations were underestimated for most of the time, indicating that calibrations in the model can still be performed.

A coarse-grid incremental pressure projection method for accelerating low Reynolds number incompressible flow simulations

Coarse grid projection (CGP) multigrid techniques are applicable to sets of equations that include at least one decoupled linear elliptic equation. In CGP, the linear elliptic equation is solved on a coarsened grid compared to the other equations, leading to savings in computations time and complexity. One of the most important applications of CGP is when a pressure correction scheme is used to obtain a numerical solution to the Navier-Stokes equations. In that case there is an elliptic pressure Poisson equation. Depending on the pressure correction scheme used, the CGP method and its performance in terms of acceleration rate and accuracy level vary. The CGP framework has been established for non-incremental pressure projection techniques. In this article, we apply CGP methodology for the first time to incremental pressure correction schemes. Both standard and rotational forms of the incremental algorithms are considered. The influence of velocity Dirichlet and natural homogenous boundary conditions in regular and irregular domains with structured and unstructured triangular finite element meshes is investigated. $L^2$ norms demonstrate that the level of accuracy of the velocity and the pressure fields is preserved for up to three levels of coarsening. For the test cases investigated, the speedup factors range from 1.248 to 102.715.

Finite element reverse time migration imaging in tunnels using an unstructured mesh

Wave field extrapolation is critical in reverse time migration (RTM). At present, wavefield extrapolation in RTM imaging for tunnels is mostly carried out via the finite difference method. However, complex tunnel models, such as those for karst and fault fracture zones, are constructed using regular grids with straight curves which can easily cause numerical dispersion and reduce imaging accuracy. In this study, the wavefield extrapolation for tunnel RTM was conducted using the finite element method, where an unstructured mesh is employed as the body-fitted partition in a complex model. The Poynting vector calculation equation suitable for the unstructured mesh finite element method was established to suppress low-frequency noise interference. The tunnel space was considered during the wavefield extrapolation to suppress mirror artefacts by using the flexibility of mesh generation. Finally, the influence of the survey layouts (one-sidewall and two-sidewall) on the tunnel imaging results was explored. The RTM results for a simple tunnel model with an inclined interface show that the method based on unstructured meshes can effectively suppress low-frequency noise and mirror artefacts, thus obtaining clearer imaging results. Also, the two-sidewall tunnel survey layout more accurately obtains the real position of the inclined interface ahead of the tunnel face. Complex tunnel numerical modelling and actual data migration results further illustrate the effectiveness of the finite element unstructured mesh method.

The inverse problem and the second order [formula omitted] scheme with finite element method used for 2D nonlinear space fractional Schrödinger equation

In this paper, we do research on the numerical analysis and the inverse problem for the two-dimensional nonlinear space fractional Schrödinger equation. The second order θ scheme combined with the unstructured mesh finite element method, which can cover the second order Crank–Nicolson method and the second order backward difference method in time, and can deal with the two-dimensional irregular region boundaries better in space, is used to obtain the numerical solution of the considered equation. On this basis, we introduce the Bayesian method to inverse the unknown parameters for the two-dimensional fractional Schrödinger equation. To testify the validity and efficiency of the proposed methods, a numerical example is conducted.